f[x_] = Exp[-x^2];
ccAnalytic = Table[Chop[N@FourierCosCoefficient[f[t], t, n], 1.*^-99], {n, 0, 20}];
ListPlot[{Log10@Abs@ccAnalytic, Sign@ccAnalytic}, Joined -> True, Mesh -> All,
PlotLegends -> LineLegend[{"Log of coefficient", "sign of coefficient"}]]